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IEEE International Conference on Trust, Security and Privacy in Computing and Communications (TrustCom)

Gray Map and Quantum Codes over the Ring F_2+uF_2+u^2F_2

Xunru Yin, Wenping Ma

Published Nov 1, 2011• 11 citations
Publisher

Abstract

In this paper, by denning the Lee distance and the Lee weight of linear codes over the ring R = F2+uF2+ u2F2, we construct a Gray map which is both an isometry and a weight-preserving map from Rnto F23n. In addition, using the proposed Gray map, we give an existence condition of quantum codes which are derived from cyclic codes over R with the Lee metric.

DOI
10.1109/trustcom.2011.122

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In this paper, by denning the Lee distance and the Lee weight of linear codes over the ring R = F2+uF2+ u2F2, we construct a Gray map which is both an isometry and a weight-preserving map from Rnto F23n. In addition, using the proposed Gray map, we give an existence condition of quantum codes which are derived from cyclic codes over R with the Lee metric.
In this study, we introduce a new Gray map which preserves the orthogonality from the chain ring F_2 [u] / (u^s ) to F^s_2 where F_2 is the finite field with two elements. We also give a condition of the existence for cyclic codes of odd length containing its dual over the ring F_2 [u] / (u^s ) . By taking advantage of this Gray map and the structure of the ring, we obtain two classes of binary quantum error correcting (QEC) codes and we finally illustrate our results by presenting some examples with good parameters.
Let R= Fq+uFq+vFq+uvFq be a commutative ring with u2=u, v2=v, uv=vu, where q is a power of an odd prime. Ashraf and Mohammad constructed some new quantum codes from cyclic codes. Under this background, another Gray map from R to Fq4 is given. This map can be naturally extended to Rn. The problem on the ring turns to the field by this isomorphic map now. Therefore, This mapping is obviously a weight-preserving and distance-preserving map. The results show that the codes after mapping are self-orthogonal codes over Fq if they are self-orthogonal codes over R. Some computational examples show that some better non-binary quantum codes can be obtained under this Gray map. We discuss the structure of linear codes. On this basis, the structure of the generating matrix of linear codes is obtained. The structure of their dual codes is also obtained. The CSS construction guarantees the existence of quantum codes. Finally, with the help of the CSS construction, we get some good quantum codes. By comparison, our quantum codes have better parameters.
On Quantum Codes Obtained From Cyclic Codes Over F2+uF2+u2F2\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2F2​+uF2​+u2F2​
Let R=F2+uF2+u2F2R=\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2R=F2​+uF2​+u2F2​ be a non-chain finite commutative ring, where u3=uu^3=uu3=u. In this paper, we mainly study the construction of quantum codes from cyclic codes over RRR. We obtained self-orthogonal codes over F2\mathbb{F}_2F2​ as gray images of linear and cyclic codes over RRR. The parameters of quantum codes which are obtained from cyclic code over RRR are discussed.